chapter 6: models for count data - macquarie university€¦ · de jong and heller glms for...

De Jong and Heller GLMs for Insurance Data Chapter 6 Solutions

Chapter 6: Models for count data

6.1 Develop a statistical model for the number of claims, in the motor vehicle insurance dataset.

Frequency distribution of numclaims:

Cumulative Cumulative

numclaims Frequency Percent Frequency Percent

--------------------------------------------------------------

0 63232 93.19 63232 93.19

1 4333 6.39 67565 99.57

2 271 0.40 67836 99.97

3 18 0.03 67854 100.00

4 2 0.00 67856 100.00

As there are so few policies with numclaims>1, it is difficult to discern a trend in plotsor tables. Figure 1 shows the number of claims (logarithmic scale) plotted against vehiclevalue, with a spline curve. As this is clearly nonlinear, we use a banded form of vehiclevalue in the model, as well as vehicle value in quadratic form. (See comments and bandingscheme in Sections 4.12 and 7.3.)

0 5 10 15 20 25 30 35

Vehicle value in $10 000 units

Num

ber

of c

laim

s

12

34

0

Figure 1: Number of claims (logarithmic scale) plotted against vehicle value, with scatterplotsmoother

Using a Poisson model with log(exposure) as offset, we find that age, area, vehicle bodyand vehicle value (banded) are all significant in single regressions. Putting them together,we get the following model selection analysis:

January 9, 2008 1


Model Deviance p AIC BIC

age 25415.33 6 25427.33 25482.08area 25491.52 6 25503.52 25558.27body 25469.41 13 25495.41 25614.04value (banded) 25485.33 6 25497.33 25552.08value 25484.60 2 25488.60 25506.85value + value2 25457.45 3 25463.45 25490.82age + area 25403.47 11 25425.47 25525.84age + body 25375.84 18 25411.84 25576.09age + value (banded) 25397.01 11 25419.01 25519.39area + body 25454.81 18 25490.81 25655.07area + value (banded) 25469.77 11 25491.77 25592.15body + value 25451.25 18 25487.25 25651.50age + body + value (banded) 25358.87 23 25404.87 25614.75age + area + body + value 25347.87 28 25403.87 25659.37age + body + value + value2 25329.61 20 25369.61 25552.11

The AIC and BIC select vastly different models, the BIC favouring model simplicity.(This effect is marked because of the large sample size.) The model with explanatoryvariables age category, vehicle body and linear and quadratic terms for vehicle value, isselected according to the AIC. SAS code and output for this model is shown. (Note thatagecat=3 as been recoded as 10 and veh body=SEDAN as ZSEDAN, to control the baselevels.)

proc genmod data=car;class agecat veh_body ;model numclaims = agecat veh_body veh_value veh_value*veh_value /

dist=poisson link=log type3 offset=logexp;run;

The GENMOD Procedure

Model Information

Data Set WORK.CAR

Distribution Poisson

Link Function Log

Dependent Variable numclaims

Offset Variable logexp

Number of Observations Read 67856

Number of Observations Used 67856

Class Level Information

Class Levels Values

agecat 6 1 2 4 5 6 10

veh_body 13 BUS CONVT COUPE HBACK HDTOP MCARA MIBUS PANVN

RDSTR STNWG TRUCK UTE ZSEDA

Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF

Deviance 68E3 25329.6101 0.3734

Scaled Deviance 68E3 25329.6101 0.3734

Pearson Chi-Square 68E3 96091.3294 1.4165

January 9, 2008 2


Scaled Pearson X2 68E3 96091.3294 1.4165

Log Likelihood -17155.7039

Algorithm converged.

Analysis Of Parameter Estimates

Standard Wald 95% Chi-

Parameter DF Estimate Error Confidence Limits Square Pr > ChiSq

Intercept 1 -2.0694 0.0539 -2.1751 -1.9638 1473.09 <.0001

agecat 1 1 0.2333 0.0528 0.1297 0.3368 19.50 <.0001

agecat 2 1 0.0587 0.0430 -0.0255 0.1430 1.87 0.1718

agecat 4 1 -0.0274 0.0412 -0.1081 0.0533 0.44 0.5062

agecat 5 1 -0.2474 0.0490 -0.3435 -0.1514 25.50 <.0001

agecat 6 1 -0.2275 0.0589 -0.3430 -0.1121 14.91 0.0001

agecat 10 0 0.0000 0.0000 0.0000 0.0000 . .

veh_body BUS 1 0.8830 0.3174 0.2610 1.5050 7.74 0.0054

veh_body CONVT 1 -0.4276 0.5871 -1.5783 0.7231 0.53 0.4664

veh_body COUPE 1 0.3943 0.1186 0.1618 0.6269 11.05 0.0009

veh_body HBACK 1 -0.0170 0.0376 -0.0907 0.0567 0.20 0.6508

veh_body HDTOP 1 0.0063 0.0902 -0.1704 0.1830 0.00 0.9445

veh_body MCARA 1 0.4371 0.2603 -0.0732 0.9473 2.82 0.0932

veh_body MIBUS 1 -0.1510 0.1515 -0.4480 0.1460 0.99 0.3190

veh_body PANVN 1 0.0409 0.1240 -0.2021 0.2839 0.11 0.7416

veh_body RDSTR 1 0.3271 0.5802 -0.8100 1.4643 0.32 0.5728

veh_body STNWG 1 -0.0736 0.0423 -0.1565 0.0094 3.02 0.0822

veh_body TRUCK 1 -0.1211 0.0922 -0.3019 0.0597 1.72 0.1892

veh_body UTE 1 -0.2641 0.0661 -0.3936 -0.1346 15.97 <.0001

veh_body ZSEDA 0 0.0000 0.0000 0.0000 0.0000 . .

veh_value 1 0.2097 0.0358 0.1396 0.2798 34.40 <.0001

veh_value*veh_value 1 -0.0234 0.0056 -0.0344 -0.0124 17.43 <.0001

Scale 0 1.0000 0.0000 1.0000 1.0000

NOTE: The scale parameter was held fixed.

LR Statistics For Type 3 Analysis

Chi-

Source DF Square Pr > ChiSq

agecat 5 89.60 <.0001

veh_body 12 42.43 <.0001

veh_value 1 45.41 <.0001

veh_value*veh_value 1 28.34 <.0001

The estimated model is y ∼ P (µ), where

ln(µ) = lnn− 2.0694+0.2333x1 + . . .− 0.2275x5 (age)+0.8830x6 + . . .− 0.2641x17 (vehicle body)+0.2097x18 − 0.0234x2

18 (vehicle value) .

• The deviance (25329.6) is well below the degrees of freedom (67856-20=67836).

• The deviance and Anscombe residuals are strongly bimodal, as shown in Figures 2and 3. The peak on the left corresponds to policies with no claims, and the bumpon the right to those with at least one claim. This is indicative of the inadequacy ofthe Poisson model. A zero–inflated Poisson model may be more appropriate. Thiscan be fitted with the gamlss software – see Chapter 10.

• The negative binomial gives an error message and strange results (κ < 0).

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Figure 2: Anscombe residuals

Figure 3: Deviance residuals

January 9, 2008 4


6.2 The SAS data file nswdeaths2002 contains all-cause mortality data for New South Wales,Australia in 2002, by age band and gender. Develop a statistical model for the number ofdeaths, using the AIC as a model selection criterion.

Death rates plotted by age and gender show a nonlinear (possibly exponential) relation-ship with age. Male death rates are higher than female death rates, at all ages. Whenlog(death rate) is plotted against age, by gender, the relationship appears linear. Also,the gender lines are roughly parallel, suggesting no age × gender interaction.

Deaths (all causes)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

<25 25-34 35-44 45-54 55-64 65-74 75-84 85+

Age group

Rat

e Male

Female

Deaths (all causes)

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0<25 25-34 35-44 45-54 55-64 65-74 75-84 85+

Age group

log

(Dea

th r

ate)

MaleFemale

Model for deaths (all causes) Start with a Poisson response distribution with loglink, and using log(population) as offset:

y ∼ P (µ) , ln µ = ln n + x′β

where y is the number of deaths due to all causes in an age-gender category.

The model with age (categorical) and gender has deviance of 261.3 on 7 d.f., indicat-ing overdispersion. (Check that no other Poisson model gives an adequate deviance.)Negative binomial model:

y ∼ NB(µ, k) , ln µ = lnn + x′β .

Models using age as a categorical covariate, and as a continuous covariate (with the mid-points of the age bands as the age values) are compared. Using the AIC, the preferredmodel has gender, age as a quadratic, and an age–gender interaction term.

Model ∆ df ` p AIC

age (cat) 16.3 8 344904.7 8 -689793.4gender 21.1 14 344872.1 2 -689740.3age (cat) + gender 17.2 7 344921.6 9 -689825.2age (cont) 16.5 14 344899.3 2 -689794.6age (cont) + gender 16.3 13 344903.8 3 -689801.6age (cont) + age (cont)2 + gender 16.8 12 344919.0 4 -689830.0age (cont) + age (cont)2 + gender +

age (cont)×gender 17.6 11 344923.0 5 -689836.0age (cont) + age (cont)2 + gender +

age (cont)×gender + age (cont)2×gender 17.6 10 344923.5 6 -689835.0

January 9, 2008 5


The fitted model is:

y

n= exp

[−7.9982 + 0.0167x1 + 0.0006x21 − 0.9004x2 + 0.0064x1x2

], k = 0.0069 .

where x1 is the midpoint of the age category and

x2 =

{1 if female0 if male

/***** Deaths (all causes) analysis: *******/data deathsnsw2002;set glm.deathsnsw2002;if age="<25" then agecont=20;if age="25-34" then agecont=30;if age="35-44" then agecont=40;if age="45-54" then agecont=50;if age="55-64" then agecont=60;if age="65-74" then agecont=70;if age="75-84" then agecont=80;if age="85+" then agecont=90;l_popn = log(popn);run;

proc genmod data=deathsnsw2002;class gender;model deaths=agecont agecont*agecont gender agecont*gender

/ dist=negbin offset=l_popn type1 type3;run;


Model Information

Data Set WORK.DEATHSNSW2002

Distribution Negative Binomial

Link Function Log

Dependent Variable Deaths Deaths

Offset Variable l_popn

Number of Observations Read 16

Number of Observations Used 16

Class Level Information

Class Levels Values

Gender 2 Female Male

Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF

Deviance 11 17.6379 1.6034

Scaled Deviance 11 17.6379 1.6034

Pearson Chi-Square 11 17.6184 1.6017

Scaled Pearson X2 11 17.6184 1.6017

Log Likelihood 344922.9857

January 9, 2008 6


Algorithm converged.

Analysis Of Parameter Estimates

Standard Wald 95% Confidence Chi-

Parameter DF Estimate Error Limits Square Pr > ChiSq

Intercept 1 -7.9882 0.1491 -8.2803 -7.6961 2872.14 <.0001

agecont 1 0.0167 0.0055 0.0058 0.0276 9.07 0.0026

agecont*agecont 1 0.0006 0.0000 0.0005 0.0007 147.57 <.0001

Gender Female 1 -0.9004 0.1220 -1.1396 -0.6613 54.44 <.0001

Gender Male 0 0.0000 0.0000 0.0000 0.0000 . .

agecont*Gender Female 1 0.0064 0.0020 0.0025 0.0103 10.50 0.0012

agecont*Gender Male 0 0.0000 0.0000 0.0000 0.0000 . .

Dispersion 1 0.0069 0.0030 0.0009 0.0128

NOTE: The negative binomial dispersion parameter was estimated by maximum likelihood.



2*Log Chi-

Source Likelihood DF Square Pr > ChiSq

Intercept 689744.154

agecont 689798.624 1 54.47 <.0001

agecont*agecont 689808.639 1 10.01 0.0016

Gender 689838.033 1 29.39 <.0001

agecont*Gender 689845.971 1 7.94 0.0048


Chi-

Source DF Square Pr > ChiSq

agecont 1 8.88 0.0029

agecont*agecont 1 37.24 <.0001

Gender 1 23.87 <.0001

agecont*Gender 1 7.94 0.0048

6.3 Develop a model for female deaths, in the Swedish mortality data set.

The negative binomial model, with orthogonal polynomials for year and age, was selectedaccording to the AIC. The minimum AIC was given by p = 28 and q = 6, and has adeviance of 7008.3 on 5927 degrees of freedom. R code is given in Chapter6Solutions.r.Plots of observed and fitted mortality are given in Figure 4.

January 9, 2008 7


1960

19701980

19902000

0

20

40

60

80

100

−10

−8

−6

−4

−2

0

Year

Age

Log

deat

h ra

te

1960

19701980

19902000

0

20

40

60

80

100

−10

−8

−6

−4

−2

0

Year

Age

Log

deat

h ra

te

Figure 4: Observed and fitted Swedish female death rates

January 9, 2008 8

chapter 6: models for count data - macquarie university€¦ · de jong and heller glms for...

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